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Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks

Adrian Celaya, Yimo Wang, David Fuentes, Beatrice Riviere

TL;DR

This work proposes two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks and uses substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

Abstract

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such as physics-informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks

TL;DR

This work proposes two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks and uses substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

Abstract

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such as physics-informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

Paper Structure

This paper contains 14 sections, 18 equations, 10 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methods
  3. Model problem
  4. DG scheme
  5. Data
  6. Network architecture
  7. Supervised approach for learning DG solutions to elliptic problems
  8. Unsupervised approach for learning DG solutions to elliptic problems
  9. Training and testing protocols
  10. Results and discussion
  11. Supervised approach
  12. Unsupervised approach
  13. Computational costs and considerations
  14. Conclusions

Figures (10)

  • Figure 1: Sketch of U-Net architecture for different network depths: $0\leq d\leq 4$. From celaya2024solutions. \newlabelfig:network0
  • Figure 1: DG solution (left), prediction (center), and absolute difference (right) for SIPG method with $\sigma = 1$ for $N = 16$, $32$, and $64$. \newlabelfig:sipg_sigma_1_results0
  • Figure 2: Setup for the function $f$ in the Darcy flow problem. In this case, $f$ is equal to $+1$ in the source region, $-1$ in the sink region, and zero everywhere else. \newlabelfig:wellschema0
  • Figure 3: From left to right, DG solution, supervised prediction, and pointwise difference for the Darcy flow problem on a $64\times 64$ grid. \newlabelfig:supwells0
  • Figure 4: From top to bottom: DG solution, test-time predictions from networks trained with 1, 10, and 100 training examples, and their absolute differences. \newlabelfig:results-num-train0
  • ...and 5 more figures